Optimal blocked minimum-support designs for non-linear models

Abstract

Finding optimal designs for experiments for non-linear models and dependent data is a challenging task. We show how the problem simplifies when the search is restricted to designs that are minimally supported; that is, the number of distinct runs (treatments) is equal to the number of unknown parameters, p, in the model. Under this restriction, the problem of finding a locally or pseudo-Bayesian D-optimal design decomposes into two simpler problems that are more widely studied. The first is that of finding a minimum-support D-optimal design d1 with p runs for the corresponding model for the mean but assuming independent observations. The second problem is finding a D-optimal block design for assigning the treatments in d1 to the experimental units. We find and assess optimal minimum-support designs for three examples, each assuming a mean model from a different member of the exponential family: binomial, Poisson and normal. In each case, the efficiencies of the designs are compared to the optimal design where the restriction on the number of distinct support points is relaxed. The optimal minimum-support designs are found to often perform satisfactorily under both local and Bayesian D-optimality for concentrated prior distributions. The results are also relatively insensitive to the assumed degree of dependence in the data.